Abstract
The electroosmotic flow of a fractional Oldroyd-B fluid in a circular microchannel is studied. The linear Navier slip velocity model is used as the chosen slip boundary condition. Exact solutions for the electric potential and transient velocity are established by means of Laplace and finite Hankel transforms. And the velocity was presented as a sum of the steady part and the unsteady one. The corresponding solutions for the fractional Maxwell fluid, fractional second grade fluid, and Newtonian fluid can also be obtained from our results. Finally, numerical results for the fluid flow are obtained and some useful conclusions are drawn. Our results may be useful for the prediction of the flow behavior of viscoelastic fluids in microchannels and can benefit the design of microfluidic devices.
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Abbreviations
- \({\bf {V}}\) :
-
Velocity vector
- R :
-
Channel radius (m)
- \(E_0\) :
-
Electric field strength (\(\hbox {Vm}^{-1}\))
- \(n_0\) :
-
Bulk ionic number concentration (\(\hbox {m}^{-3}\))
- u :
-
Electroosmotic flow velocity in z direction (\(\hbox {ms}^{-1}\))
- d :
-
Slip length (m)
- \(z_{\mathrm v}\) :
-
Valence of ions
- e :
-
Electron charge (\(1.602\times 10^{-19}\,\hbox {C}\))
- T :
-
Absolute temperature (K)
- K :
-
Dimensionless electrokinetic width
- \(I_0(\cdot )\) :
-
Modified Bessel function of the first kind
- \(J_0(\cdot )\) :
-
Bessel function of the first kind
- \(\tau\) :
-
Shear stress (Pa)
- \(\gamma\) :
-
Shear strain (rad)
- \(\mu\) :
-
Viscosity constant (Pa s)
- \(\lambda\) :
-
Relaxation time (s)
- \(\theta\) :
-
Retardation time (s)
- \(\alpha\) :
-
Order of fractional derivative
- \(\beta\) :
-
Order of fractional derivative
- \(\varepsilon\) :
-
Dielectric constant
- \(\psi\) :
-
Potential distribution (V)
- \(\psi _\omega\) :
-
Zeta potential of the channel wall (V)
- \(\rho\) :
-
Density of the electrolyte solution (\(\hbox {kg}\,\hbox {m}^{-3}\))
- \(\rho _{\mathrm e}\) :
-
Electric charge density (\(\hbox {Cm}^{-3}\))
- \(\kappa _{\mathrm{B}}\) :
-
Boltzmann constant (\(1.381\times 10^{-23}\,\hbox {JK}^{-1}\))
- \(\kappa\) :
-
Debye–Hückel parameter (\(\hbox {m}^{-1}\))
- \(\beta _m\) :
-
Positive roots of Eq. (24)
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11672163 and 11472161), the Natural Science Foundation of Shandong Province, China (Grant Nos. ZR2015AM011 and ZR2014AQ015), and the Independent Innovation Foundation of Shandong University, China (Grant No. 2013ZRYQ002).
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This article is part of the topical collection “2016 International Conference of Microfluidics, Nanofluidics and Lab-on-a-Chip, Dalian, China” guest edited by Chun Yang, Carolyn Ren and Xiangchun Xuan.
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Jiang, Y., Qi, H., Xu, H. et al. Transient electroosmotic slip flow of fractional Oldroyd-B fluids. Microfluid Nanofluid 21, 7 (2017). https://doi.org/10.1007/s10404-016-1843-x
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DOI: https://doi.org/10.1007/s10404-016-1843-x